Topic 11 - Topic 11 ANOVA II Balanced Two-Way ANOVA...

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1 Topic 11 – ANOVA II Balanced Two-Way ANOVA (Chapter 19)
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2 Two Way ANOVA We are now interested the combined effects of two factors, A and B, on a response (note: text refers to these as R = rows and C = columns – we’ll call them A, B, and later C for a 3-way ANOVA). Examples: Want to consider the effects of diet plan (Factor A) and exercise program (Factor B) on weight. Want to consider the effects of a drug (Factor A) and a vitamin tablet (Factor B) on blood pressure.
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3 Two Way ANOVA (2) Interested in a combination of the two factors; unlike blocking, both of primary interest. Could treat each combination of factors as a treatment and do one-way ANOVA, but then you have to use contrasts a lot in order to tests hypotheses of interest.
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4 Two Way ANOVA (3) Interaction is a possibility. Replication is required to investigate interaction. You generally want at least two observations per treatment combination. You also generally want a balanced design (for this topic, we’ll assume cell sizes are equal).
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5 Example (Problem 19.1) An animal experiment is designed to investigate whether the drug Levorphanol reduces stress as reflected in the cortical sterone level. It is also likely that Epinephrine (adrenaline) levels have some effect and there may be an interaction with the drug effect as well. Some animals were given a drug that raised their normal levels of Epinephrine.
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6 Example (2) Control Levorphanol Epinephrine Both 1.90 1.80 1.54 4.10 1.89 0.82 3.36 1.64 1.74 1.21 5.33 4.84 5.26 4.92 6.07 3.08 1.42 4.54 1.25 2.57
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7 Example (3) If we treat this as one-way ANOVA (which is not ideal in real life but useful here for instructional purposes), then we can design contrasts can be used to investigate the effects: Comparison C L E B L effect E effect Interaction -1 -1 1 1 -1 -1 -1 1 -1 1 1 1
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8 SAS Code (one-way) proc glm ; class trt; model y=trt; contrast 'L' trt - 1 1 - 1 1 ; contrast 'E' trt - 1 - 1 1 1 ; contrast 'Interaction' trt 1 - 1 - 1 1 ; means trt / tukey ; run ; quit ;
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9 SAS Output Source DF SS MS F Value Pr > F Model 3 37.58 12.53 12.30 0.0002 Error 16 16.30 1.02 Total 19 53.88 Contrast DF SS MS F Value Pr > F L 1 12.83 12.83 12.60 0.0027 E 1 18.59 18.59 18.25 0.0006 Interaction 1 6.16 6.16 6.05 0.0257
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10 Notes The three contrasts have a special property – they are “orthogonal”. Their sum is actually the model SS: 12.83 + 18.59 + 6.16 = 37.58. We see that there are distinguishable effects for both drugs, plus an interaction (F-tests). A significant interaction means that the size of the L effect is different at different levels of E (or equivalently, the size of the E effect is different for different levels of L); more later.
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11 Two-Way ANOVA Break up the treatments into two factors. Factor 1:
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Topic 11 - Topic 11 ANOVA II Balanced Two-Way ANOVA...

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